TY - GEN
T1 - Regularized covariance estimation in scaled Gaussian models
AU - Wiesel, Ami
PY - 2011
Y1 - 2011
N2 - We consider regularized covariance estimation in scaled Gaussian settings, e.g., Elliptical distributions, compound-Gaussian processes and spherically invariant random vectors. The classical maximum likelihood (ML) estimate due to Tyler is asymptotically optimal under different criteria and can be efficiently computed even though the optimization is non-convex. We propose a unified framework for regularizing this estimate in order to improve its finite sample performance. Our approach is based on the discovery of hidden convexity within the ML objective, namely convexity on the manifold of positive definite matrices. We regularize the problem using appropriately convex penalties. These allow for shrinkage towards the identity matrix, shrinkage towards a diagonal matrix, shrinkage towards a given positive definite matrix, and regularization of the condition number. We demonstrate the advantages of these estimators using numerical simulations.
AB - We consider regularized covariance estimation in scaled Gaussian settings, e.g., Elliptical distributions, compound-Gaussian processes and spherically invariant random vectors. The classical maximum likelihood (ML) estimate due to Tyler is asymptotically optimal under different criteria and can be efficiently computed even though the optimization is non-convex. We propose a unified framework for regularizing this estimate in order to improve its finite sample performance. Our approach is based on the discovery of hidden convexity within the ML objective, namely convexity on the manifold of positive definite matrices. We regularize the problem using appropriately convex penalties. These allow for shrinkage towards the identity matrix, shrinkage towards a diagonal matrix, shrinkage towards a given positive definite matrix, and regularization of the condition number. We demonstrate the advantages of these estimators using numerical simulations.
KW - Covariance estimation
KW - hidden convexity
KW - optimization on manifolds
KW - regularization
KW - robust statistics
UR - http://www.scopus.com/inward/record.url?scp=84857184729&partnerID=8YFLogxK
U2 - 10.1109/CAMSAP.2011.6136012
DO - 10.1109/CAMSAP.2011.6136012
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AN - SCOPUS:84857184729
SN - 9781457721052
T3 - 2011 4th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2011
SP - 309
EP - 312
BT - 2011 4th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2011
T2 - 2011 4th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2011
Y2 - 13 December 2011 through 16 December 2011
ER -